Meanwhile, when the two eigenvalues differ \(\epsilon _2>\epsilon _1\)  and the eigenvectors are real, the refractive index varies between its largest value \(\sqrt{\epsilon _2}\)  (propagation along \({e}_{1}\) ) and its smallest value \(\sqrt{\epsilon _1}\)  (propagation along \({e}_{2}\) ). The angle dependence of the refractive index \({\rm n}(\theta )\)  now either traces out an ellipse (Fig. REF b), when \(\epsilon _1\)  and \(\epsilon _2\)  are both positive, or a hyperbola (Fig. REF c), when \(\epsilon _1\)  and \(\epsilon _2\)  have different signs, constituting a hyperbolic materialSee [1]} for details about these fascinating materials.. At the transition between elliptical and hyperbolic dispersion, the smallest eigenvalue passes through zero \(\epsilon _1\rightarrow 0\) . In this case the eccentricity of the dispersion ellipse tends to unity, and the ellipse is compressed into a line, as shown in Fig. REF b. Again the refractive index vanishes, but now this only occurs for one direction of propagation. For instance taking \({e}_{1}={e}_{x}\)  and \({e}_{2}={e}_{y}\)  we have,
\({\rm n}(\theta )=\sqrt{\frac{\epsilon _1\epsilon _2}{\epsilon _2\sin ^2(\theta )+\epsilon _1\cos ^2(\theta )}}={\left\lbrace \begin{array}{ll}\sqrt{\epsilon _2}\rightarrow 0&\; (\text{Propagation along }x)\\\sqrt{\epsilon _1}\ne 0&\; (\text{Propagation along }y).\end{array}\right.}\) 
